High-fidelity time-evolving simulations of a rigid parachute trailing behind a descent module in a supersonic flight regime have been performed, employing Large-Eddy Simulation (LES) and Immersed-Boundary Method (IBM) techniques. The study aims to establish the fluid dynamic nature of the ‘breathing’ instability present also in a rigid decelerator, and thus its independence from structural flexibility. The turbulent wake of the descent capsule interacts with the bow shock generated by the parachute acting as the primary triggering factor. Energetic turbulent structures, accurately resolved by Large-Eddy Simulation, induce local fluctuations in the parachute shock, destabilizing its equilibrium with the upstream flow and leading to continuous cyclic motion of the shock wave. This motion correlates with periodic variations in flow pressure inside the canopy control volume, impacting parachute performance. Based on simulation results, a zero-dimensional model is developed to predict the unsteady dynamics of the shock motion and the decelerator performance. The model is driven by input fluctuations from the capsule wake, reproducing the main frequencies of shock position oscillations and drag variations as observed in simulations. It is apparent that unsteadiness is eventually triggered by low-frequency wake perturbations. Thus, the study provides insights into factors contributing to unsteady parachute responses in supersonic regimes.